To find the vertex, focus, and directrix of the parabola given by the equation [tex]\( x^2 - 12x = 8y - 92 \)[/tex], we need to rewrite this equation in the standard form of a parabola. Let's go through the steps in detail:
1. Rewrite the given equation in standard parabola form:
[tex]\[
x^2 - 12x = 8y - 92
\][/tex]
2. Completing the square for the [tex]\( x \)[/tex]-terms:
- Start with [tex]\( x^2 - 12x \)[/tex].
- To complete the square, take half of the coefficient of [tex]\( x \)[/tex], which is [tex]\( \frac{-12}{2} = -6 \)[/tex]. Then square it to get [tex]\( (-6)^2 = 36 \)[/tex].
- Add and subtract this square inside the equation:
[tex]\[
x^2 - 12x + 36 - 36 = 8y - 92
\][/tex]
- This simplifies to:
[tex]\[
(x - 6)^2 - 36 = 8y - 92
\][/tex]
3. Move the constant terms to the right side of the equation:
- Add 36 to both sides:
[tex]\[
(x - 6)^2 = 8y - 92 + 36
\][/tex]
- Simplify the right side:
[tex]\[
(x - 6)^2 = 8y - 56
\][/tex]
4. Rewrite the equation in standard form [tex]\( (x - h)^2 = 4p(y - k) \)[/tex]:
- By comparing it to the standard form, we identify that [tex]\( h = 6 \)[/tex], [tex]\( k = 7 \)[/tex], and [tex]\( 4p = 8 \)[/tex].
- Solve for [tex]\( p \)[/tex] by dividing both sides by 4:
[tex]\[
p = \frac{8}{4} = 2
\][/tex]
Thus, the vertex (h, k) of the parabola is [tex]\((6, 7)\)[/tex].
5. Determining the focus:
- The focus is located at [tex]\((h, k + p)\)[/tex].
- Substituting [tex]\( h = 6 \)[/tex], [tex]\( k = 7 \)[/tex], and [tex]\( p = 2 \)[/tex], we get:
[tex]\[
\text{Focus} = (6, 7 + 2) = (6, 9)
\][/tex]
6. Determining the directrix:
- The directrix is the line [tex]\( y = k - p \)[/tex].
- Substituting [tex]\( k = 7 \)[/tex] and [tex]\( p = 2 \)[/tex], we get:
[tex]\[
\text{Directrix} = y = 7 - 2 = 5
\][/tex]
Final Result:
- Vertex: [tex]\((6, 7)\)[/tex]
- Focus: [tex]\((6, 9)\)[/tex]
- Directrix: [tex]\(y = 5\)[/tex]
Therefore, the correct answer is:
[tex]\(\text{D. vertex:} (6, 7) \text{ focus:} (6, 9) \text{ directrix:} y=5 \)[/tex]
